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Intuitively, Bianchi identities state that "the boundary of a boundary is zero".
For example, let's consider a disk:
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The disk has a boundary, which is a circle. A circle has no boundary.
Next, let's consider a line segment:
It is one-dimensional like the circle but has a boundary: the two endpoints. But then again, this boundary (=the two endpoints) don't have a boundary.
When we get to gauge theories we will see that Maxwell's equations are a special case of the Yang-Mills equations, which describe not only electromagnetism but also the strong and weak nuclear forces. A generalization of the identity $d^2=0$, the Bianchi identity, implies conservation of "charge" in all these theories - although these theories have different kinds of "charge". Similarly, we will see when we get to general relativity that due to the Bianchi identity, Einstein's equations for gravity automatically imply local conservation of energy and momentum!
page 96 in Gauge fields, knots, and gravity by John Baez
The boundary of a boundary is zero John Wheeler
The Bianchi identities have a close connection the Noether's second theorem.